General definition of a discriminant

Does the discriminant have any specific definition? I've come across two discriminant definitions that don't seem to be very relevant to each other, and I was wondering if there is any particular property that discriminants convey. In finding roots and using the quadratic equation, I've seen ## D = b^2 - 4ac## and for multivariable calculus,I've seen ## D = f_{xx}f_{yy} - (f_{xy})^2##. These two equations don't seem very similar, yet are called discriminants (to my knowledge). Is there any single definition for a discriminant in mathematics? Does it determine anything specific about functions? Are there more definitions for discriminants? I realize these are just terms for different operations but it seemed odd they have the same names.

Does the discriminant have any specific definition? I've come across two discriminant definitions that don't seem to be very relevant to each other, and I was wondering if there is any particular property that discriminants convey. In finding roots and using the quadratic equation, I've seen ## D = b^2 - 4ac## and for multivariable calculus,I've seen ## D = f_{xx}f_{yy} - (f_{xy})^2##. These two equations don't seem very similar, yet are called discriminants (to my knowledge). Is there any single definition for a discriminant in mathematics? Does it determine anything specific about functions? Are there more definitions for discriminants? I realize these are just terms for different operations but it seemed odd they have the same names.

It's rare that a single term is used in the same way across the sciences. For example, the term 'vector' means one thing in mathematics, quite another when talking biology.

In the quadratic formula, the evaluation of the discriminant determines if the roots of the quadratic are 1) real and distinct, 2) real and identical, or 3) complex conjugates.

This is an article which talks about some of the mathematical uses of the term 'discriminant':